Multiplicative noise removal is a challenging problem in image denoising.In this paper,we develop a nonlocal matrix rank minimization method for the multiplicative noise removal problem.By utilizing the logarithm tran...Multiplicative noise removal is a challenging problem in image denoising.In this paper,we develop a nonlocal matrix rank minimization method for the multiplicative noise removal problem.By utilizing the logarithm transformation,we convert the problem into an additive noise removal problem and propose a maximum a posteriori(MAP)estimation-based matrix rank minimization model for this kind of additive noise removal.A proximal alternating algorithm is designed to solve the matrix rank minimization model.The convergence of the algorithm is demonstrated by the famous Kurdyka-Lojasiewicz property.Taking advantage of the proposed matrix rank minimization model and its proximal alternating algorithm,a multiplicative noise removal method is finally developed.Numerical experiments illustrate that the proposed method can remove multiplicative noise in images much better than the existing state-of-the-art methods in terms of both image recovered measure quantities and visual qualities.展开更多
In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new re...In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.展开更多
In this paper, the authors discuss the relationship in detail between the rank of M in the modified matrix M = A ＋ BC^＊ and the rank of matrix A. The authors do believe the results are useful tools in the modified m...In this paper, the authors discuss the relationship in detail between the rank of M in the modified matrix M = A ＋ BC^＊ and the rank of matrix A. The authors do believe the results are useful tools in the modified matrices.展开更多
The method of recovering a low-rank matrix with an unknown fraction whose entries are arbitrarily corrupted is known as the robust principal component analysis （RPCA）. This RPCA problem, under some conditions, can b...The method of recovering a low-rank matrix with an unknown fraction whose entries are arbitrarily corrupted is known as the robust principal component analysis （RPCA）. This RPCA problem, under some conditions, can be exactly solved via convex optimization by minimizing a combination of the nuclear norm and the 11 norm. In this paper, an algorithm based on the Douglas-Rachford splitting method is proposed for solving the RPCA problem. First, the convex optimization problem is solved by canceling the constraint of the variables, and ~hen the proximity operators of the objective function are computed alternately. The new algorithm can exactly recover the low-rank and sparse components simultaneously, and it is proved to be convergent. Numerical simulations demonstrate the practical utility of the proposed algorithm.展开更多
This paper gives the rank of matrix and equalities and inequalities of the difference number of non-zero eigenvalue, and discuss the equivalent description of multi angle of equalities for upper and lower bounds of th...This paper gives the rank of matrix and equalities and inequalities of the difference number of non-zero eigenvalue, and discuss the equivalent description of multi angle of equalities for upper and lower bounds of the inequality.展开更多
<span style="line-height:1.5;"><span>In this paper, we consider a constrained low rank approximation problem: </span><img src="Edit_57d85c54-7822-4512-aafc-f0b0295a8f75.png" wi...<span style="line-height:1.5;"><span>In this paper, we consider a constrained low rank approximation problem: </span><img src="Edit_57d85c54-7822-4512-aafc-f0b0295a8f75.png" width="100" height="24" alt="" /></span><span style="line-height:1.5;"><span>, where </span><i><span>E</span></i><span> is a given complex matrix, </span><i><span>p</span></i><span> is a positive integer, and </span></span><span style="line-height:1.5;"></span><span style="line-height:1.5;"><span> is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation </span><img src="Edit_ced08299-d2dc-4dbb-907a-4d8d36d2e87a.png" width="60" height="16" alt="" /></span><span style="line-height:1.5;"><span>. We discuss the range of </span><i><span>p</span></i><span> and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions. And an algorithm for the problem is proposed and the numerical example is given to show its feasibility.展开更多
A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with...A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with the recovery of fully perturbed low-rank matrices. By utilizing the p-null space property (p-NSP) and the p-restricted isometry property (p-RIP) of the matrix, sufficient conditions to ensure that the stable and accurate reconstruction for low-rank matrix in the case of full perturbation are derived, and two upper bound recovery error estimation ns are given. These estimations are characterized by two vital aspects, one involving the best r-approximation error and the other concerning the overall noise. Specifically, this paper obtains two new error upper bounds based on the fact that p-RIP and p-NSP are able to recover accurately and stably low-rank matrix, and to some extent improve the conditions corresponding to RIP.展开更多
随着基于位置社交网络(location-based social network,LBSN)的发展,兴趣点推荐成为满足用户个性化需求、减轻信息过载问题的重要手段.然而,已有的兴趣点推荐算法存在如下的问题:1)多数已有的兴趣点推荐算法简化用户签到频率数据,仅使...随着基于位置社交网络(location-based social network,LBSN)的发展,兴趣点推荐成为满足用户个性化需求、减轻信息过载问题的重要手段.然而,已有的兴趣点推荐算法存在如下的问题:1)多数已有的兴趣点推荐算法简化用户签到频率数据,仅使用二进制值来表示用户是否访问一个兴趣点;2)基于矩阵分解的兴趣点推荐算法把签到频率数据和传统推荐系统中的评分数据等同看待,使用高斯分布模型建模用户的签到行为;3)忽视用户签到数据的隐式反馈属性.为解决以上问题,提出一个基于Ranking的泊松矩阵分解兴趣点推荐算法.首先,根据LBSN中用户的签到行为特点,利用泊松分布模型替代高斯分布模型建模用户在兴趣点上签到行为;然后采用BPR(Bayesian personalized ranking)标准优化泊松矩阵分解的损失函数,拟合用户在兴趣点对上的偏序关系;最后,利用包含地域影响力的正则化因子约束泊松矩阵分解的过程.在真实数据集上的实验结果表明:基于Ranking的泊松矩阵分解兴趣点推荐算法的性能优于传统的兴趣点推荐算法.展开更多
The paper researches the rank of combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.Using the properties of the idempotent matrix and elementary block matrix operation,we get some rank equalities for combi...The paper researches the rank of combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.Using the properties of the idempotent matrix and elementary block matrix operation,we get some rank equalities for combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.These rank equalities generalize the results of Koliha J J,Rakoevi V and Tian Y,and give some applications of the rank equalities.展开更多
The rank of a graph is defined to be the rank of its adjacency matrix. In this paper, the Matlab was used to explore the graphs with rank no more than 5;the performance of the proposed method was compared with former ...The rank of a graph is defined to be the rank of its adjacency matrix. In this paper, the Matlab was used to explore the graphs with rank no more than 5;the performance of the proposed method was compared with former methods, which is simpler and clearer;and the results show that all graphs with rank no more than 5 are characterized.展开更多
An eigenvector method for ranking alternatives whose measurements are given as vague values is provided. Firstly, a positive matrix is constructed which is defined as evaluation information matrix （EIM）. Based on fo...An eigenvector method for ranking alternatives whose measurements are given as vague values is provided. Firstly, a positive matrix is constructed which is defined as evaluation information matrix （EIM）. Based on four assumptions for evaluating alternatives, a ranking eigenvector is defined. And then it is proved, based on positive matrix theory, that the EIM＇s eigenvector corresponding to the maximal eigenvalue is the ranking vector. For alternatives whose characteristics are presented by vague sets, the proposed techniques can evaluate the degree of suitability to which an alternative satisfies the decision-maker＇ s requirement efficiently.展开更多
In this paper we study the perturbation bound of the projection ( W A ) ( W A )+,where both the matrices A and W are given with W positive diagonal and severely stiff.When the perturbed matrix (A)= A + δA satisfy sev...In this paper we study the perturbation bound of the projection ( W A ) ( W A )+,where both the matrices A and W are given with W positive diagonal and severely stiff.When the perturbed matrix (A)= A + δA satisfy several row rank preserving conditions,we derive a new perturbation bound of the projection.展开更多
With respect to the multiple attribute decision making problems with linguistic preference relations on alternatives in the form of incomplete linguistic judgment matrix, a method is proposed to analyze the decision p...With respect to the multiple attribute decision making problems with linguistic preference relations on alternatives in the form of incomplete linguistic judgment matrix, a method is proposed to analyze the decision problem. The incomplete linguistic judgment matrix is transformed into incomplete fuzzy judgment matrix and an optimization model is developed on the basis of incomplete fuzzy judgment matrix provided by the decision maker and the decision matrix to determine attribute weights by Lagrange multiplier method. Then the overall values of all alternatives are calculated to rank them. A numerical example is given to illustrate the feasibility and practicality of the proposed method.展开更多
Principal Component Analysis (PCA) is a widely used technique for data analysis and dimensionality reduction, but its sensitivity to feature scale and outliers limits its applicability. Robust Principal Component Anal...Principal Component Analysis (PCA) is a widely used technique for data analysis and dimensionality reduction, but its sensitivity to feature scale and outliers limits its applicability. Robust Principal Component Analysis (RPCA) addresses these limitations by decomposing data into a low-rank matrix capturing the underlying structure and a sparse matrix identifying outliers, enhancing robustness against noise and outliers. This paper introduces a novel RPCA variant, Robust PCA Integrating Sparse and Low-rank Priors (RPCA-SL). Each prior targets a specific aspect of the data’s underlying structure and their combination allows for a more nuanced and accurate separation of the main data components from outliers and noise. Then RPCA-SL is solved by employing a proximal gradient algorithm for improved anomaly detection and data decomposition. Experimental results on simulation and real data demonstrate significant advancements.展开更多
In this paper a non-iterative technique is developed for the correction of faulty antenna array based on matrix pencil technique（MPT）. The failure of a sensor in antenna array can damage the radiation power pattern ...In this paper a non-iterative technique is developed for the correction of faulty antenna array based on matrix pencil technique（MPT）. The failure of a sensor in antenna array can damage the radiation power pattern in terms of sidelobes level and nulls. In the developed technique, the radiation pattern of the array is sampled to form discrete power pattern information set. Then this information set can be arranged in the form of Hankel matrix（HM） and execute the singular value decomposition（SVD）. By removing nonprincipal values, we obtain an optimum lower rank estimation of HM. This lower rank matrix corresponds to the corrected pattern. Then the proposed technique is employed to recover the weight excitation and position allocations from the estimated matrix. Numerical simulations confirm the efficiency of the proposed technique, which is compared with the available techniques in terms of sidelobes level and nulls.展开更多
Generally, the least-squares problem can be solved by the normal equation. Based on the projection theorem, we propose a direct method to investigate the maximal and minimal ranks and inertias of the least-squares sol...Generally, the least-squares problem can be solved by the normal equation. Based on the projection theorem, we propose a direct method to investigate the maximal and minimal ranks and inertias of the least-squares solutions of matrix equation AXB = C under Hermitian constraint, and the corresponding formulas for calculating the rank and inertia are derived.展开更多
基金the researchers for providing the Matlab codes of the comparison methods.
文摘Multiplicative noise removal is a challenging problem in image denoising.In this paper,we develop a nonlocal matrix rank minimization method for the multiplicative noise removal problem.By utilizing the logarithm transformation,we convert the problem into an additive noise removal problem and propose a maximum a posteriori(MAP)estimation-based matrix rank minimization model for this kind of additive noise removal.A proximal alternating algorithm is designed to solve the matrix rank minimization model.The convergence of the algorithm is demonstrated by the famous Kurdyka-Lojasiewicz property.Taking advantage of the proposed matrix rank minimization model and its proximal alternating algorithm,a multiplicative noise removal method is finally developed.Numerical experiments illustrate that the proposed method can remove multiplicative noise in images much better than the existing state-of-the-art methods in terms of both image recovered measure quantities and visual qualities.
基金supported by the National Natural Science Foundation of China (No. 91320201 and No. 61471262)the International (Regional) Collaborative Key Research Projects (No. 61520106002)
基金Project supported by the National Natural Science Foundation of China （Grant No.60672160）
文摘In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.
基金the National Natural Sciences Foundation of China(10371044)the Science and Technology Commission of Shanghai Municipality through Grant(04JC14031)+1 种基金the University Young Teacher Sciences Foundation of Anhui Province(2006jq1220zd)Supported by the Ph.D.,Program Scholarship Fund of ECNU(2007)
文摘In this paper, the authors discuss the relationship in detail between the rank of M in the modified matrix M = A ＋ BC^＊ and the rank of matrix A. The authors do believe the results are useful tools in the modified matrices.
基金supported by the National Natural Science Foundation of China(No.61271014)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20124301110003)the Graduated Students Innovation Fund of Hunan Province(No.CX2012B238)
文摘The method of recovering a low-rank matrix with an unknown fraction whose entries are arbitrarily corrupted is known as the robust principal component analysis （RPCA）. This RPCA problem, under some conditions, can be exactly solved via convex optimization by minimizing a combination of the nuclear norm and the 11 norm. In this paper, an algorithm based on the Douglas-Rachford splitting method is proposed for solving the RPCA problem. First, the convex optimization problem is solved by canceling the constraint of the variables, and ~hen the proximity operators of the objective function are computed alternately. The new algorithm can exactly recover the low-rank and sparse components simultaneously, and it is proved to be convergent. Numerical simulations demonstrate the practical utility of the proposed algorithm.
文摘This paper gives the rank of matrix and equalities and inequalities of the difference number of non-zero eigenvalue, and discuss the equivalent description of multi angle of equalities for upper and lower bounds of the inequality.
文摘<span style="line-height:1.5;"><span>In this paper, we consider a constrained low rank approximation problem: </span><img src="Edit_57d85c54-7822-4512-aafc-f0b0295a8f75.png" width="100" height="24" alt="" /></span><span style="line-height:1.5;"><span>, where </span><i><span>E</span></i><span> is a given complex matrix, </span><i><span>p</span></i><span> is a positive integer, and </span></span><span style="line-height:1.5;"></span><span style="line-height:1.5;"><span> is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation </span><img src="Edit_ced08299-d2dc-4dbb-907a-4d8d36d2e87a.png" width="60" height="16" alt="" /></span><span style="line-height:1.5;"><span>. We discuss the range of </span><i><span>p</span></i><span> and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions. And an algorithm for the problem is proposed and the numerical example is given to show its feasibility.
文摘A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with the recovery of fully perturbed low-rank matrices. By utilizing the p-null space property (p-NSP) and the p-restricted isometry property (p-RIP) of the matrix, sufficient conditions to ensure that the stable and accurate reconstruction for low-rank matrix in the case of full perturbation are derived, and two upper bound recovery error estimation ns are given. These estimations are characterized by two vital aspects, one involving the best r-approximation error and the other concerning the overall noise. Specifically, this paper obtains two new error upper bounds based on the fact that p-RIP and p-NSP are able to recover accurately and stably low-rank matrix, and to some extent improve the conditions corresponding to RIP.
文摘随着基于位置社交网络(location-based social network,LBSN)的发展,兴趣点推荐成为满足用户个性化需求、减轻信息过载问题的重要手段.然而,已有的兴趣点推荐算法存在如下的问题:1)多数已有的兴趣点推荐算法简化用户签到频率数据,仅使用二进制值来表示用户是否访问一个兴趣点;2)基于矩阵分解的兴趣点推荐算法把签到频率数据和传统推荐系统中的评分数据等同看待,使用高斯分布模型建模用户的签到行为;3)忽视用户签到数据的隐式反馈属性.为解决以上问题,提出一个基于Ranking的泊松矩阵分解兴趣点推荐算法.首先,根据LBSN中用户的签到行为特点,利用泊松分布模型替代高斯分布模型建模用户在兴趣点上签到行为;然后采用BPR(Bayesian personalized ranking)标准优化泊松矩阵分解的损失函数,拟合用户在兴趣点对上的偏序关系;最后,利用包含地域影响力的正则化因子约束泊松矩阵分解的过程.在真实数据集上的实验结果表明:基于Ranking的泊松矩阵分解兴趣点推荐算法的性能优于传统的兴趣点推荐算法.
文摘The paper researches the rank of combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.Using the properties of the idempotent matrix and elementary block matrix operation,we get some rank equalities for combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.These rank equalities generalize the results of Koliha J J,Rakoevi V and Tian Y,and give some applications of the rank equalities.
文摘The rank of a graph is defined to be the rank of its adjacency matrix. In this paper, the Matlab was used to explore the graphs with rank no more than 5;the performance of the proposed method was compared with former methods, which is simpler and clearer;and the results show that all graphs with rank no more than 5 are characterized.
基金Sponsored by the Basic Research Foundation of Beijing Institute of Technology(BIT-UBF-20070842009)
文摘An eigenvector method for ranking alternatives whose measurements are given as vague values is provided. Firstly, a positive matrix is constructed which is defined as evaluation information matrix （EIM）. Based on four assumptions for evaluating alternatives, a ranking eigenvector is defined. And then it is proved, based on positive matrix theory, that the EIM＇s eigenvector corresponding to the maximal eigenvalue is the ranking vector. For alternatives whose characteristics are presented by vague sets, the proposed techniques can evaluate the degree of suitability to which an alternative satisfies the decision-maker＇ s requirement efficiently.
文摘In this paper we study the perturbation bound of the projection ( W A ) ( W A )+,where both the matrices A and W are given with W positive diagonal and severely stiff.When the perturbed matrix (A)= A + δA satisfy several row rank preserving conditions,we derive a new perturbation bound of the projection.
基金the National Natural Science Foundation of China (70701008)National Science Foundationfor Distinguished Young Scholars of China (70525002)
文摘With respect to the multiple attribute decision making problems with linguistic preference relations on alternatives in the form of incomplete linguistic judgment matrix, a method is proposed to analyze the decision problem. The incomplete linguistic judgment matrix is transformed into incomplete fuzzy judgment matrix and an optimization model is developed on the basis of incomplete fuzzy judgment matrix provided by the decision maker and the decision matrix to determine attribute weights by Lagrange multiplier method. Then the overall values of all alternatives are calculated to rank them. A numerical example is given to illustrate the feasibility and practicality of the proposed method.
文摘Principal Component Analysis (PCA) is a widely used technique for data analysis and dimensionality reduction, but its sensitivity to feature scale and outliers limits its applicability. Robust Principal Component Analysis (RPCA) addresses these limitations by decomposing data into a low-rank matrix capturing the underlying structure and a sparse matrix identifying outliers, enhancing robustness against noise and outliers. This paper introduces a novel RPCA variant, Robust PCA Integrating Sparse and Low-rank Priors (RPCA-SL). Each prior targets a specific aspect of the data’s underlying structure and their combination allows for a more nuanced and accurate separation of the main data components from outliers and noise. Then RPCA-SL is solved by employing a proximal gradient algorithm for improved anomaly detection and data decomposition. Experimental results on simulation and real data demonstrate significant advancements.
基金sypported by the Research Management Centre(RMC),School of Postgraduate Studies(SPS),Communication Engineering Department,Faculty of Electrical Engineering(FKE),Universiti Teknologi Malaysia(UTM),Johor Bahru(Grant Nos.12H09 and 03E20)
文摘In this paper a non-iterative technique is developed for the correction of faulty antenna array based on matrix pencil technique（MPT）. The failure of a sensor in antenna array can damage the radiation power pattern in terms of sidelobes level and nulls. In the developed technique, the radiation pattern of the array is sampled to form discrete power pattern information set. Then this information set can be arranged in the form of Hankel matrix（HM） and execute the singular value decomposition（SVD）. By removing nonprincipal values, we obtain an optimum lower rank estimation of HM. This lower rank matrix corresponds to the corrected pattern. Then the proposed technique is employed to recover the weight excitation and position allocations from the estimated matrix. Numerical simulations confirm the efficiency of the proposed technique, which is compared with the available techniques in terms of sidelobes level and nulls.
基金Supported by the Science Foundation Project of Tianshui Normal University(TSA1315)
文摘Generally, the least-squares problem can be solved by the normal equation. Based on the projection theorem, we propose a direct method to investigate the maximal and minimal ranks and inertias of the least-squares solutions of matrix equation AXB = C under Hermitian constraint, and the corresponding formulas for calculating the rank and inertia are derived.
